Use with the (group-theory) tag. The tag "finite-groups" refers to questions asked in the field of Group Theory which, in particular, focus on the groups of finite order.

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1answer
23 views

Normal subgroup $N$, subgroup $U$, then $UN/N = U/N$.

Let $G$ be a group and $N \unlhd G$ a normal subgroup, $U \le G$ some subgroup. Then I guess $U / N$ is always some group, and moreover $U / N = UN / N$, because $UN / N = \{ unN : u \in U, n \in N \} ...
6
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1answer
77 views

If $G$ is solvable and $G/[G,G]$ is cyclic, can $G\times G$ be generated by 2 elements?

I doubt this is true, but I haven't found any small counterexamples (there are no counterexamples with $|G| < 1536, |G| \neq 768$): Suppose $G$ is finite, solvable, 2-generated, and $G/[G,G]$ ...
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1answer
22 views

Normal Sylow $p$-subgroup of a normal subgroup

Any hints for the following question - I am sure that I am missing something very simple here. $K$ is a normal subgroup of $G$ and $P$ is a Sylow $p$-subgroup of $K$. If $P$ is a normal subgroup of ...
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0answers
38 views

Existence of a map $\phi:\mathbb{Z}_{N^2}^* \mapsto \mathbb{F} $

Is there a map between the group of $\mathbb{Z}_{N^2}^*$ where $N$ is a composite number , a product of two equal size secure prime numbers $p$ and $q$ and a finite field $\mathbb{F}$, such that for ...
1
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1answer
24 views

Splitting field of an irreducible polynomial of degree five $f \in \mathbb{Q}[X]$

Let $\Omega/\mathbb{Q}$ be the splitting field of an irreducible polynomial $f \in Q[X]$ of degree five. Show that $Gal(\Omega/\mathbb{Q})$ equals $A_5$ or $S_5$ if $Gal(\Omega/\mathbb{Q})$ has an ...
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2answers
26 views

Group Theory (Abstract Algebra) question! o(G) vs o(g)??

If G is a group, and g is an element of G, what is the difference between the following two notations o(G) and o(g)?
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1answer
34 views

How does the base of a group determine the “sort” of the elements in the group

I'm trying to study groups in Mathematica, and I've asked a question on Mathematica.SE that perhaps only someone from Math.SE could answer. Related: How does ...
0
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1answer
30 views

$p$-subgroups conjugate iff $\cong$ to Sylow p-subgroups of some other groups?

Let $G$ be a finite group and $p$ a prime such that $p^\alpha$ divides $|G|$ and $p^{\alpha+1} \nmid |G|$. I know that Sylow $p$-subgroups of $G$ are conjugate to one another but if we have some ...
1
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1answer
17 views

Simple group question need help…

Alright so I've got a question here in terms of groups. So define $\omega = {e}^{2i\pi\over 13}$ -The exponent of e should be $2i\pi\over 13$ but it's not coming clear when as an exponent of e there ...
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3answers
69 views

Can we find some constraint about order of $xy$ in a group $G$?

Can we determine order of $xy$ in $G$ if we know order of $x$ and $y$ ? I know that answer is yes for abelian groups and I guess the answer is no for nonabelian case. That is why I am lookking for ...
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0answers
30 views

Do normal group endomorphisms form a normal submonoid?

What it says on the tin. A group endomorphism $v\colon G\to G$ is called normal if $v(aba^{-1})=av(b)a^{-1}$ for all $a,b\in G$. Equivalently, the map $g\mapsto v(g^{-1})g$ is a group homomorphism. ...
0
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0answers
34 views

Semidirect products of cyclic groups

Consider $A=\langle a\rangle$, cyclic group of order $9$ and $B=\langle b\rangle$, cyclic group of order $3$. Consider now the following action of $B$ on $A$ via automorphism: ...
0
votes
1answer
51 views

Subgroups of direct products

Consider a group $G$ which is a direct product of two groups of coprime order: $G = G_1 \times G_2$ with $|G_1|=n_1$, $|G_2|=n_2$ and $\textrm{gcd}(n_1, n_2)=1$. Let $H \le G$. Is it true that ...
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4answers
46 views

Show that $G$ is a group if the cancellation law holds when identity element is not sure to be in $G$

Having already read through show-that-g-is-a-group-if-g-is-finite-the-operation-is-associative-and-cancel, however in Herstein's Abstract Algebra, I was required to prove it when we're not sure if ...
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2answers
39 views

Find $b \in G$ for each $a \in G$, such that $b^m = a$, where $m$ is coprime to the order of group G

Let $(G, \cdot)$ be a finite group (with identity element $1 \in G$) and $m \in \mathbb{N}$ be coprime to $|G|$. Proofe that for all $a \in G$ there exists exactly one $b \in G$, such that $b^m = a$. ...
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2answers
138 views

Why free presentations ? Why not permutation or matrix representations?

Two days ago, I asked why free presentations? and frankly I did not get a convincing answer. I am trying here to ask the question in a different way : We know that a group can be defined by a ...
2
votes
2answers
79 views

How many subgroups of $\Bbb{Z}_4 \times \Bbb{Z}_6$?

I have been trying to calculate the number of subgroups of the direct cross product $\Bbb{Z}_4 \times \Bbb{Z}_6.$ Using Goursat's Theorem, I can calculate 16. Here's the info: Goursat's Theorem: Let ...
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2answers
27 views

The Direct Product of Groups and their Subgroups

Let $G_1$, $G_2$ be groups of prime power order. Write $|G_1|=p^m$ and $|G_2|=q^n$ for some $0 \leq m,n$. (The primes $p,q$ need not be distinct.) Let $H_1$ be a subgroup of $G_1$ and let $H_2$ ...
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1answer
121 views

Subgroups of abelian $p$-groups

Let $A$ be an Abelian group of prime power order. It can be expressed as a (unique) direct product of cyclic groups of prime power order: $A = \mathbb{Z}_{p^{n_1}} \times \cdots \times ...
4
votes
2answers
123 views

Show that $G$ ( subgroup of $\mathrm{GL}(E)$) is finite.

I came across with, I think, a difficult problem : Let E a Hermitian space with a Hermitian norm $||\ ||$. We provide $\mathcal{L}(E)$ with the norm $|||\ \ |||$ subordinated to $||\ ||$. ...
0
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0answers
22 views

Sylow subgroups and normalizers [duplicate]

Let $H$ be a Sylow 3-subgroup and $K$ a Sylow 5-subgroup of a finite group $G$. Suppose $|H|=3$ and $|K|=5$ and $N_G(K)$, has an element of order 3. Show that $N_G(H)$ has an element of order 5.
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0answers
26 views

$ K \rtimes_{\phi_1} C \cong K \rtimes_{\phi_2} C$ when $\phi_1(C), \phi_2(C)$ are conjugated and $C$ is a product of two cyclic groups

I know the following result: Let $C$ be a finite cyclic group, $K$ a finite group such that there exist homomorphisms $\phi_1,\phi_2$ $\phi_i:C \to Aut(K) $ such that $\phi_1(C), \phi_2(C)$ are ...
0
votes
1answer
37 views

Is this object a group?

$\begin{array}{ccccccccc} \times&e_0&e_1&e_2&e_3&e_4&e_5&e_6&e_7\\ e_0&I&e_1&e_2&e_3&e_4&e_5&e_6&e_7\\ ...
5
votes
1answer
80 views

Does $\mathrm{GL}_{n-2}(\mathbb{Z})$ has an element of order $m$?

Let me introduce the context: A few week ago I have made the following contest as a "homework" : ENS contest (France) 2006 which is essentially about $SL_n(\mathbb{Z})$ group, finite subgroup of ...
0
votes
1answer
23 views

Summing the traces of matrix powers

Let $G=\langle h\rangle_n\subset{\rm GL}(m,\mathbb{C})$ be a cyclic group of order $n$. I wonder if there is a good formula for calculating the sum $\sum_{g\in G}{\rm Tr}(g)$ via ${\rm Tr}(h)$, for ...
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1answer
21 views

Can't remember definition of $\lvert G \rvert_{p'}$

For $G$ a finite group, I know that $\lvert G\rvert$ denotes the order of the group. My question is: What is $\lvert G\rvert_{p'}$? Also is this the same as $\lvert G\rvert_p$ (without the prime on ...
0
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1answer
15 views

What is the formula for products in dihedral groups?

Let $G \colon = \langle x, y \ | \ x^2 = y^n = e, \ x^iy^j = x^{i^{\prime}} y^{j^{\prime}} \ $ if and only if $ i = i^{\prime}, j = j^{\prime}, \ xy = y^{-1}x \rangle$. That is, let $G$ be the ...
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0answers
21 views

Let $G$ be a group of order 24 and suppose $n_2(G) > 1 \ \ and \ \ n_3(G) > 1$ . Then $G \cong S_4$

My attempt is : Since $n_3 > 1$ and $n_3 \equiv 1 \ \ mod \ \ 3 $ and divides 8, then the only possibilty is $n_3 = 4$ and thus $| G:N| = 4$, where $N = N_G(P)$ and $P \in Syl_3(G)$. Then $G/K $ ...
4
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2answers
103 views

Why free presentations?

What is the motivation to study "free" presentations of groups,even though all (or almost all) the questions (or the problems) concerning this type of presentations are known to be undecidable ?
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2answers
74 views

$g\in G$ maximal order in $G$ abelian then $G=\left<g\right>\oplus H$

If $g\in G$ is an element of maximal order in a finite abelian group $G$ then exists $H\leq G$ such that $G=\left<g\right>\oplus H$ Attempt: Using fundamental theorem I know that ...
0
votes
1answer
50 views

Subgroups of finite Abelian groups

I am interested in finding all of the subgroups (up to isomorphism) of a finite Abelian group $A$. I know the following: -- A finite Abelian group $A$ can be represented as a direct product of ...
3
votes
1answer
32 views

Having trouble grasping the class equation as an explanation as to why a conjugate class's order divides the order of a group.

Suppose $|G|$ is a prime power $p^n$ and that $N$ is a normal subgroup of $G$. Show that $|y^G|$ is a power of $p$ whenever $y \in G$ Attempt: Firstly, I assume that $y^G = \{ gyg^{-1} | g \in G ...
1
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1answer
33 views

Number of degree-$d$ representations of a perfect group?

It seems to be a standard result that the number of degree-1 representations of a group $G$ is equal to $[G : G']$ where $G'$ is the commutator subgroup (e.g. Lemma 6.2.7 in the 2012 textbook ...
3
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1answer
41 views

Property of isomorphic subgroups in finite groups

I have the following question: Does there exist a finite group $G$ and two subgroups $U,H\leq G$, s.t. the following properties are satisfied: a) $H\cong U$. b) There is no subgroup $L$, s.t. ...
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1answer
31 views

$2$-Sylow subgroups of $S_{14}$

I need to describe what are the $2$-Sylow subgroups of $S_{14}$. I don't know how to start thinking about this since is it too large. I know that all that I must do is to find one of them since all ...
0
votes
1answer
14 views

Show that the orbit of $H$ containing $x$ is equal to the right coset $Hx$

Let $G$ be a group and let $H$ be a subgroup of $G$. Let $X$ be the set of elements of $G$. Let $ \ast : H \times X \to X$ be given by $$ h \ast x = hx (h \in H, x \in X)$$. QUESTION: Let $x \in X$. ...
2
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2answers
88 views

group theory proof

Let $G=\langle t \rangle$ be a cyclic group with $\text{ord}(t)=n$. I want to show that for all $d|n$ it holds that $$\left\lbrace s \in G ;\text{ord}(s)=d \right\rbrace=\left\lbrace t^{\frac{n}{d}k} ...
1
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1answer
23 views

Does this condition gaurantee the cyclicity of a finite abelian group?

Let $G$ be a finite abelian group in which there are at most $n$ solutions of the equation $x^n = e$ for each posivite integer $n$. How to determine if $G$ is cyclic or not?
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2answers
32 views

Product of cyclic groups

How can you quickly tell that the product of cyclic groups $\mathbb{Z}_4 \times \mathbb{Z}_3$ has a 2-subgroup containing an element of order 4? Also, I don't understand the notion of multiplying ...
0
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1answer
44 views

Which elements of this cyclic group would generate it?

Let $n$ be a given arbitrary positive integer, and let $U_n$ denote the group of all the positive integers less than $n$ and relatively prime to $n$ under multiplication mod $n$. Then for which values ...
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1answer
33 views

What about the index of this subgroup? [duplicate]

Let $G$ be a group, and let $H$ be a subgroup of finite index in $G$, and let $N \colon = \cap_{x \in G} \ xHx^{-1}$. Then $N$ is clearly a subgroup of $G$ which is contained in $H$ and such that ...
3
votes
3answers
82 views

Question about soluble and cyclic groups of order pq

I'm trying to solve the following problem: If $p>q$ are prime, show that a group $g$ of order $pq$ is soluble. If $q$ doesn't divide $p-1$, show that $pq$ is cyclic. Show that two non abelian ...
3
votes
0answers
45 views

How to solve this problem on finite groups? [duplicate]

Let $G$ be a finite group whose order is not divisible by $3$ and such that $(ab)^3 = a^3 b^3$ for all $a$, $b$ in $G$. Then can we determine if $G$ is abelian or not? Since $$ (ab)^3 = a^3 b^3 $$ ...
0
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1answer
22 views

Determinantal order of character of a group.

The notion of determinantal order can be found in 'Character Theory of finite groups' by I Martin Isaacs. If $\chi$ be a linear character of a finite group G, show that the order of $\chi$ in the ...
3
votes
1answer
101 views

Order and element set of the group with presentation $\langle a,b \;|\; a^9=1, b^3=a^3, [a,b]=a^3\rangle$

If we have a group presentation $G=\langle a,b \;|\; a^9=1, b^3=a^3, [a,b]=a^3\rangle$, how we will get the following values: The order of the group. The elements of the group written in terms of ...
1
vote
1answer
41 views

Building a partial injective relation

Question : A Partial Injective Relation from $A \rightarrow B$ is maximal if its graph of an injection function from $A$ to $B$ or the graph of an injection function from $B$ to $A$. Example: $A ...
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vote
1answer
78 views

Can someone please explain the word problem (from group theory) in Calculus III layman's terms

I'd appreciate it if someone could offer a schematic explanation of the word problem from group theory in terms which someone at a calculus III level (but without any background in group theory or ...
2
votes
1answer
18 views

Permutations, cycles and conjugacy

Let $u \in S_n$ be a cycle, where $S_n$ is the group of permutations of the set with $n$ elements. Let $\sigma \in S_n$ such that the support of $\sigma \circ u \circ \sigma^{-1}$ is the same as the ...
2
votes
0answers
27 views

Rational representation of a 136 order group

Let G be the group of order 136 = 8 * 17 with presentation $$<x,y : y^{8}=x^{17}=1 \quad yxy^{-1}=x^{4} >$$ Find the simple summands of the group ring Q[G].
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6answers
205 views

The use of conjugacy class and centralizer?

This is more or less for a conceptual and better-understanding question in group theory and in representation theory: (1) Why are conjugacy class and centralizer important concepts in the group / ...